1.1 Graphing Exercise: Production Possibilities Curve

The economizing problem — scarce resources and unlimited wants — highlights the need for society to make choices among available alternatives. The production possibilities curve is a graphical illustration of the options that are available at a given point in time.

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Exploration: Production possibilities: what alternatives are available to society?

The first graph illustrates the amounts of pizza and industrial robots that can be produced with a hypothetical society's currently available resources and technology. Each point on the curve represents the greatest number of robots that society can produce if it chooses to produce the corresponding quantity of pizza. At any point on the curve, producing more pizzas means fewer robots can be produced. Likewise, producing more robots means less pizza can be produced. The graph shows that, initially, society is choosing to produce 2 units of pizza (200,000 pizzas) and 7 units of industrial robots (7,000 robots). To use the graph, drag the blue triangle on the Pizza axis to the left or right to change the production mix and investigate opportunity costs. Clicking again on the triangle will establish that as a starting point. Click the Reset button to start over.

1. If society produces 200,000 pizzas (2 units—point "C" on the graph), what is the greatest number of robots that can be produced?
   See answer here
2. Starting from 2 units of pizza, what is the opportunity cost of the third unit of pizzas?
   See answer here
3. What happens to the opportunity cost of pizzas as even more pizzas are produced?
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4. What happens to the opportunity cost of robots as robot production is increased?
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1.2 Graphing Exercise: Present Choices and Future Possibilities

The amount of resources available to an economy at some future point depends upon the choices it makes today. As one simple example, if a country experiences a loss of population—whether by emigration, the devastation of war, or a reduction in the number of births, fewer (human) resources will be available in the future to produce goods and services. This exercise will investigate the extent to which an economy's current production choices affect its future production capabilities.

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Exploration: How do present choices affect future possibilities?

To use the graph, use the mouse to drag the scroll bar button to the left or right, observing the impact that different current production choices have on the future position of the production possibilities curve.

1. How must society choose to produce today if it wishes its economy to grow faster?
   See answer here
2. What is the opportunity cost of faster economic growth?
   See answer here

1.3 Graphing Exercise: Curves and Slopes, Part 1—Straight Lines

Economics is the study of relationships—between wants and resources, between price and quantity, between consumption and income, just to name a few. As you will discover throughout the text, many of these relationships can be presented graphically. Graphs can be a handy tool to summarize quickly the relationships between two variables.

If the two variables of interest increase or decrease together, the relationship is said to be direct and the graph of the relationship will be an upward sloping line. If the two variables move in opposite directions, the relationship is said to be inverse and the graph of the relationship will be a downward sloping line.

In many cases, these relationships can reasonably be represented by a straight line. The slope of the line is the ratio of the vertical change to the horizontal change between any two points on the line. Generally, the slope tells us the amount by which one variable changes when the other variable increases by one unit. The general equation of a straight line is y = a + bx, where:

y is the value of the variable on the vertical axis (the dependent variable),
x is the value of the variable on the horizontal axis (the independent variable),
a is the vertical intercept, the value at which the line crosses the vertical axis, and
b is the slope

(The exception is for a vertical line, for which the slope is infinite and there is no vertical intercept.)

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Exploration: How are lines, slopes, and intercepts related?

The position of any straight line is determined by two points. The graphing applet will allow you to graph straight lines by clicking anywhere in the graph to establish an initial point and clicking again to establish a second point. (x and y values are displayed above the graph to help in locating points.) Click the Plot Equation button to draw a line connecting the points. The accompanying table will list several points along the line; the vertical intercept is always given by the value of y corresponding to a zero value of x. Finally, the equation of the line is displayed above the graph.

1. Click on the point x = 0, y = 50. (The location of points can be abbreviated (x, y) where x is the value of x and y is the value of y.) Next, click on the point (10, 0) and plot the resulting line. What is the equation for this line? Does it represent a direct or an inverse relationship? What is the vertical intercept? What is the slope?
   See answer here
2. Click Reset, then click on the points (3, 30) and (6, 60). Plot the line. What is the equation for this line? Does it represent a direct or an inverse relationship? What is the vertical intercept? What is the slope?
   See answer here
3. Graph a horizontal line with a vertical intercept of 40. What is the equation for this line? What is its slope?
   See answer here
4. Click Reset. Click on the point (2, 20). Move your mouse such that x increases by 2 and y increases by 10; click on this point. What is the equation for the line? What is the vertical intercept and slope? Verify that the slope measures the ratio of the vertical change to the horizontal change of your two points.
   See answer here
5. Use the values in the table to verify that the slope is the same between any two points along this line.
   See answer here

1.3 Graphing Exercise: Curves and Slopes, Part 2—Curved Lines

Not all economic relationships are best represented by straight lines. For example, as production of an item increases, its per-unit cost usually falls at first, then increases. As another example, taxes paid do not usually increase in proportion to income. Instead, taxes tend to rise faster and faster as income increases.

In these and many other cases, we are still interested in the slope of the line representing the relationship—at any particular point, if one variable changes, how much will the other variable change? For example, consider any particular level of income. We might like to know how much taxes will rise if income increases starting from that particular point. Or, we might like to know by how much average cost changes as output changes from its current level. Unlike a straight line whose slope is always the same at any point on the line, the slope of a curved line will vary as we move along the curve. The slope at any point is best represented by the slope of a straight line that is tangent to the curve.

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Exploration: How do we find the slope of a nonlinear curve?

The graphing applet will calculate the slope of a nonlinear curve by drawing in a straight line tangent to the graph. To use the graph, click and drag the blue diamond to the left or right—the slope of the line is calculated and displayed in the box. Click the Open Up/Down button to change the shape of the curve.

1. Click on the Open Up/Down button until the curve is open at the bottom. What is the slope of the curved line when X is 1?
   See answer here
2. What happens to the slope of the line as X increases from a value of 1 to 3?
   See answer here
3. What is the slope of the curved line when X is 5? How do you interpret the minus sign?
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4. What happens to the slope of the line as X increases from a value of 5 to a value of 7?
   See answer here
5. At what X value does the line have a zero slope?
   See answer here
6. Experiment on your own: If the curve is getting "flatter," is the slope rising or falling? What can you say about positive and negative slopes?
   See answer here
7. Click on the Open Up/Down button until the curve is open at the top. Repeat the previous questions for this graph.
   See answer here
8. Comparing the "open down" and "open up" curves, speculate as to how knowledge of the slope—and how it is changing—might indicate whether a graph opens upward or downward.
   See answer here